Constrained elastica under edge thrust: Nonlinear springy walls

The post-buckling behavior of a slender beam that is laterally constrained between two parallel walls is studied, where one wall is fixed and the other is pushed by the beam against a nonlinear spring. This model system is of relevance to a range of engineering applications and physical systems, such as deep drilling, stent procedures, and filopodia growth in living cells. The mathematical model accounts for large rotation and for various possible contact scenarios between the beam and the walls. The predictions of the model are compared with published experiments and show very good qualitative and quantitative agreement, thus reconciling important discrepancies between the results of published small-deformation models and available experimental observations. Although the large-deformation analysis cannot provide closed-form analytical description, we were able to identify key features of the behavior by careful examination of the mathematical structure of the nonlinear governing equations. In particular, we provide universal maps related to the occurrence of critical events during the loading process, which are expressed in terms of two non-dimensional quantities that describe the mode-corrected relative stiffness and relative gap between the walls. These universal maps enable a simple and straight-forward characterization of the system behavior, in all modes, and therefore provide an excellent platform for engineering design and for practical use. Based on these universal maps, one is able to describe, almost completely, the expected behavior of the system. In addition, extensive numerical results demonstrate the richness of possible behavior, and how it depends on all parameters of the system in an intricate manner.